A definable relation between c.e. sets and ideals

The Pi-0-1 classes have become important structures in computability theory. Related to the study of properties of individual classes is the study of the lattice of all Pi-0-1 classes, denoted E_Pi. We define a substructure of E_Pi, G = [N, 2^omega] for N nonprincipal, and a quotient structure of G, denoted G^diamond and thought of as G modulo principal classes disjoint from N. Using the setting of computably enumerable ideals, we present basic results for G and G^diamond and show that G^diamond is isomorphic to E*, the lattice of computably enumerable sets modulo finite difference. This isomorphism allows us to transfer invariant classes from E* to E_Pi. However, it does not in general allow the transfer of orbits. We give the conditions under which an orbit could transfer and an example of one which does, and conclude with open questions related to degree-theoretic properties.